Theory of Surface Spin Waves in the Heisenberg Ferromagnet

The eigenvalue matrix for a number of semi-infinite cubic system of spins coupled by nearest neighbors is derived for the Heisenberg ferromagnet. The mathematical formulation used allows solutions for an arbitrary variation in the surface exchange constants. Surface spin-wave modes for the simple cubic structure are calculated for the {100} and {110} surfaces. By allowing the exchange parameters to deviate from the bulk values, the full richness of the surface spin-wave spectra is displayed. Surface spin-wave modes are found to exist above and below the bulk spectrum. The dispersion curves can be truncated, i.e., they exist only over a limited region of ${\mathrm{k}}_{11}$ space, where ${\mathrm{k}}_{11}$ is the two-dimensional propagation vector parallel to the crystal surface. The eigenvectors of the surface spin-wave modes are also discussed.

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